inﬂuence of crosslinking on micromechanical ... · inﬂuence of crosslinking on micromechanical...

Bulgarian Chemical Communications, Volume 47, Special Issue B (pp. 433–441) 2015

Influence of crosslinking on micromechanical characteristics ofliquid silicone rubber. Numerical simulations of microindentation process

G. Zamfirova1∗, S. Cherneva2, V. Gaydarov1, T. Vladkova3

1 Todor Kableshkov University of Transport, 158 Geo Milev Str., BG-1576 Sofia, Bulgaria2 Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, BG-1113 Sofia, Bulgaria

3 University of Chemical Technology and Metallurgy, 8 Kliment Ohridski Blvd., BG-1756 Sofia, Bulgaria

Mechanical properties of three samples from liquid silicone rubber were investigated by means of microindentation experiments andfinite-element simulations. The experimental and numerical load-displacement curves were compared and showed good coincidence.One of the aims of present work was seeking of correlation between some standard classical mechanical characteristics and parametersobtained by micro- and nanoindentation. The influence of the crosslinking on mechanical properties was investigated as well. Moreoverparameters describing plastic behavior of investigated materials (such as yield strength, distribution of equivalent Von Mises stress ininvestigated materials during the process of microindentation, distribution of the equivalent plastic strain after unloading), which aren’tpossible to be obtained only by means of microindentaion experiments, were determined by means of numerical simulations.

Key words: liquid silicone rubber, microindentation, mechanical properties, numerical simulations, finite-element method

INTRODUCTION

Investigations of mechanical properties of poly-mer materials are very important, because they giveinformation about the technological and exploitationcharacteristics of the materials and their applicabil-ity for different purposes. From classical mechan-ical investigations (for example tensile test) at con-stant deformation rate or creep test at constant load,or whatever static or dynamic mechanical experiment,we could not obtain direct information about thesupramolecular structure because the standard me-chanical approaches evaluate the material as a whole,including all types of its defects. Unfortunately,the complexity of the molecular and supramolecu-lar structure of the polymers as well as the struc-tural imperfections, macro-, micro- and nanodefects,the relations between standard mechanical propertiesand structure are difficult to be determined and inter-preted. The increasing information and experience inthe field of micromechanical methods give opportu-nity to consider the micro- and nano-mechanical in-vestigations as adequate tools for elucidating somestructural peculiarities of polymeric materials.

The aim of present work is to look for corre-lation between some standard classical mechanicalcharacteristics and parameters obtained by micro- andnanoindentation. This could be considered as a linkbetween structural peculiarities of the sample and itsmacromechanical behaviour.

∗ To whom all correspondence should be sent:[email protected]

EXPERIMENTAL

Materials

Three silicone rubber samples with differentcrosslinking were investigated. They are commercialproduct of KCC Corporation, Korea from the groupLiquid Silicone Rubber (LSR) consisting mainly ofsilicone polymer and fumed silica. Liquid siliconerubber is cured automatically in liquid injection mold-ing machines.

The components are supplied at a regular ratio.This mixed compound is measured by volume usingsome mechanical tools in the injection unit. Finallyheat curing is done very quickly in a hot mold. Afterthe fixed cure time, de-molding is completed and thecured articles are taken from the mold. In addition,the cure system includes curing by platunum catalystso therefore, it is physiologically inert.

Mechanical properties of investigated silicon rub-ber materials, provided by producer are shown in theTable 1.

It is obviously from Table 1 that from sample 1to sample 3 Shore hardness increases and elongation

Table 1. Mechanical properties of the three types of sili-cone rubber provided by the manufacturer

Tensile TearSample Hardness strength Elongation strength

(Shore A) (MPa) (%) (N/mm)

1. SL7240 39 8 750 362. SL7250 49 10 600 273. SL7270 67 9 400 40

© 2015 Bulgarian Academy of Sciences, Union of Chemists in Bulgaria 433

G. Zamfirova et al.: Influence of crosslinking on micromechanical characteristics of liquid silicone rubber. Numerical simulations of...

decreases. There is inverse proportion between Shorehardness and elongation which is usually attributedto increasing the cross-linking. So the increasing thesample number means the increasing of the degree ofcross-linking. Tensile strength is the maximum stressthat a material can withstand while being stretched orpulled before failing or breaking.

Tear strength is a specific characteristic which iswidely used when study mechanical behavior of rub-ber or textile materials. It is defined as a maximumstress obtained from stress-strain experiment dividedto sample thickness. There is some similarity betweenthe tear strength and yield strength what concerns toinitiation of plastic deformation but loading test andsample geometry are different.

Tensile strength and tear strength are not influ-enced linearly by degree of crosslinking.

Methods

This investigation was carried out predominantlyby method known as a depth-sensing indentation(DSI) or instrumented indentation testing (IIT). Themethod consists of recording by a testing device at aconstant loading speed the magnitude of the force asa function of penetration depth for each point on theloading (or unloading). Large amount of mechanicalparameters are determined using indentation curves,obtained by above mentioned method:

— Dynamic hardness (DH)

HD =aFh2 , (1)

where (F) is the value of the instant load at loadingand unloading testing regime, (a = 3.8584) is a con-stant which depends on the shape of the indenter and(h) is an indentation depth. This characteristic revealshow the material responds to plastic, elastic and vis-coelastic deformation during the test.

— Martens hardness (HMs) is determined fromthe slope (m) of the increasing load-displacementcurve in the 50% ÷ 90% Fmax interval and charac-terizes the material resistance of penetration:

HMs =1

26.43m2 . (2)

This characteristic has similar physical sense as dy-namic hardness, but characterizes material propertiesat maximum indentation depth and constant load.

— Indentation hardness (Hit) is determined usingOliver-Pharr approximation method and measure re-

sistance to permanent deformation.

Hit =Fmax

24.50h2c, (3)

where (hc) is the depth of contact of the indenter withthe test sample.

— Indentation Elastic Modulus (Eit) is calculatedfrom unloading part of the load-displacement curve:

1Er

=1−ν2

s

Eit+

1−ν2i

Ei, (4)

where (Er) is the experimentally converted elasticmodulus, based on indentation contact, (νs) is thePoisson’s ratio of specimen, whereas (Ei) and (νi) arethe Young’s modulus and Poisson’s ratio for indenter,respectively.

— Indentation creep (Cit) which is a relativechange in the indentation depth at constant test forceis calculated as:

Cit =h2−h2

h1, (5)

where (h1) and (h2) are indentation depths at the be-ginning and at the end of the creep measurement.

— Elastic part of indentation work (ηit) is deter-mined from the areas under loaded and unloaded partof the load- unload test (W =

∫Pdh):

ηit =Wel

Wel−Wpl. (6)

Vickers hardness, HV*, is obtained as a functionof computed indentation hardness. Its direct measure-ment from the diagonal of the residual imprint wasimpossible because of the absence of residual imprint.

For more comprehensive characterization of thesamples, we apply hybrid experimental-numerical ap-proach to the indentation experiment, which com-bines microindentation experiments with numeri-cal simulations by means of finite-element method(FEM). The combination of the indentation experi-ment and its numerical simulation has a number ofadvantages, because enables determination of somevaluable mechanical properties (yield strength, distri-bution of equivalent Von Mises stress, distribution ofthe equivalent plastic and elastic strain in the zone un-der indenter and etc.).

The experimental conditions, which we used forconducting microindentation tests, are following:

— First mode is loading- unloading;• Indenter is Vickers diamond pyramid with an-

gle of 136◦;

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• Loading rate of 0.0250 mN/s;• Maximum load – 0.8 mN;• Minimum load – 0.020 mN;• Holding time after unloading 0 s;• Poisson’s ratio 0.47;• Indenter Poisson’s ratio of 0.07;• Indenter Modulus of 1140 GPaMeasurements have been carried out at room tem-

perature. At least 30 indentations for each samplehave been made and average value and the meansquare error have been calculated, which could beconsidered as an indicator of the microstructural het-erogeneity of the samples.

Also a measuring by the mode loading-holding atmaximum load-unloading with holding time 60 s fordetermining the indentation creep has been made. Allother experimental conditions are the same.

The equilibrium swelling and effective crosslinkdensity were determined by Dogadkin-Tarrasova’smethod using toluene as a swelling agent [1].

Numerical simulations of microindentation process

Boundary value problem. The boundary value prob-lem is defined under the following assumptions:• The indentation process is quasistatic;• The deformable axisymmetric specimen is

composed by an isotropic linear elastic-plasticwith linear hardening material;• Normally indenter’s material is very hard (usu-

ally this is a material with elastic modulus ≈103 GPa). For that reason it is accepted, thatthe indenter can be modelled as a rigid body;• The friction forces in the contact area are ne-

glected;• No stress-strain prehistory is taken into ac-

count.The equation of motion of the deformable body is:

σi j, j = 0, (7)

where σi j is the Cauchy stress tensor. Because of theaxial symmetry the boundary condition on G1 is (seeFig. 1):

uy|G1 = 0. (8)

The deformable body is imposed on a rigid baseand this gives the following boundary condition onthe surface G5:

ux|G5= 0. (9)

The indenter I penetrates from the side of contactsurface G2 with a prescribed velocity νI . The depth

Fig. 1. Scheme of the boundary value problem.

of the contact zone l2 during the deformation processchanges. On the contact zone the velocity is given as:

ν|G2= νI, (10)

Material model. The material model used in presentwork is elastic-plastic with linear hardening. Themodel involves the following material parameters:Young’s modulus, Poisson’s ratio, yield strength,strength coefficient.

The strain tensor εi j is used to describe the defor-mation in a material point belonging to the investi-gated material:

εi j =12(ui, j +u j,i) , (11)

where {u1,u2,u3} is the displacement vector of thematerial point at position {x1,x2,x3} in the referenceconfiguration. The material model applied to investi-gated materials is described below.

The total strain tensor εi j is a sum of the elasticstrain tensor εe

i j and the plastic strain tensor εpi j:

εi j = εei j + ε

pi j. (12)

The elastic strain tensor is related to the stress ten-sor through Hooke’s law:

εei j =

1+ν

Eδi j +

1−2ν

3Eσklδklδi j, (13)

where E is the Young’s modulus and ν is the Poisson’sratio of the particular material.

Following the von Mises yield criterion, yieldingoccurs in a point when the next validates:

F (σi j,εp)≡ 3

2si jsi j−σ

2p (ε

p) = 0, (14)

where si j is deviator of the stresses and εp is the

equivalent plastic strain. The flow stress σp of thematerial depends on the accumulated plastic strain

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and the value of the initial yield stress σ0.2p . If the

initial yield stress is exceeded, the material starts toshow plastic yielding. It is assumed that the materialis isotropic with linear hardening behavior. In thiscase the evolution of σp during plastic deformation isgiven by:

σp (εp) = σ

0.2p +Kε

p, (15)

where K is the strength coefficient.The accumulated equivalent plastic strain is given

by:

εp =

∫ t

0

√23

.ε

pi j

.ε

pi jdt (16)

According to the associated plastic flow rule, theplastic strain rate is given by

.ε

pi j =

.

λ∂F∂σi j

, (17)

where.

λ is the plastic multiplier.

Indentation test procedure modelling. The processof indenter penetration and separation from the spec-imen is simulated as a contact problem. It is evidentfrom the experiment that the deformation caused bythe indenter is located in a small area around the in-denter. For that reason a boundary value problem isdefined in a cylindrical domain around the indenter’stip with sufficiently large radius compared to the sizeof the indenter. The radius is chosen in such way thatthe deformation due to the penetration vanishes at thelateral and the bottom of the considered cylindricaldomain.

The friction forces on the contact surface are ne-glected. The experiment that has to be simulated nu-merically is a microindentation, where the indenteris a tetrahedral Vickers pyramid with tip angle 136◦.Because of geometrical symmetry, the considered do-main can be reduced and only one fourth of the wholedomain is used for solving the boundary value prob-lem. Often the problem is simplified by substitutingthe geometry of the Vickers pyramid with the circu-lar cone with an angle of 70.3◦ between the axis andthe generatrix. Such approach has been applied forexample in [2–4]. In this case the problem can beconsidered as an axisymmetric one. The geometry ofthe 2D model is shown in Fig. 1.

Numerical simulations. The above described pro-cess of microindentation is modelled numerically.The finite- element model has been developed using

the finite-element code MSC.MARC [5]. The bound-ary value problem is based on the model described inSections 2.3.1, 2.3.2 and 2.3.3. The geometry of thedomain is given on Fig. 1: l1=0.5 mm, l4=2 mm.

The relation between l3 and l4 is l3l4= 1

20 . In thisway the influence of boundary conditions on the nu-merical solution is avoided [6].

As initial values for the material properties of in-vestigated materials, we used data taken from the cer-tificates of these materials, available at the Web pageof the producer (values are shown in Table 1) and lit-erature [7]. They are given in second column of Ta-ble 3. For the modelling process of microindentation,3 series of finite element simulations with 3600 four-nodes isoparametric finite elements with full integra-tion have been performed (Fig. 2). The experimen-tal and calculated load-displacement curves were thencompared. In this way we performed a trial-error pro-cedure in order to obtain the material parameter setthat gives the best fit to the experimental data. Theyare shown on third column of Table 3.

Fig. 2. Enlargement part from FE mesh for the 2D modelaround the indenter.

RESULTS AND DISCUSSIONS

Fig. 3 shows experimental indentation curves forthree silicon rubber samples with different degreesof crosslinking. The loading and unloading parts ofindentation curves are close to each other, which istypical for materials with high elasticity. Indentationcurve for sample 2 is the most inclined, i.e. it possesthe smallest hardness. This can be seen directly fromthe diagram in the Fig.4, which illustrate the values ofdynamic hardness (DH), Martens hardness (MHs), in-dentation hardness (Hit) and Vickers hardness (HV*).

It is obviously from Figs. 3 and 4 that all micro-hardness characteristics are not directly dependent on

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Fig. 3. Experimental indentation curves for three siliconrubber samples with different degrees of crosslinking.

Fig. 4. All hardness characteristics for the three samples.

degree of crosslinking. The same is valid for indenta-tion modulus and elastic part of the indentation work.

No relation has been found between the mi-croindentation parameters and tensile strength, whichcould be expected because microindentation param-eters characterize the local mechanical propertieswhile the tensile strength is statistically measured pa-rameter. It depends not only on material propertiesbut also on existing of macro-defects in loaded zone,which become stress concentrators and increase untilthe break of the material.

The dependence of hardness characteristics fromtear strength is shown on Fig. 5a. It is observed al-most linear dependence. Moreover both characteris-tics which are related to resistance against plastic de-formation (Hit and HV*) are more sensitive to tearstrength. Dynamic and Martens hardnesses whichcharacterize the resistance against total deformation,including elastic and plastic deformation componentsare less sensitive to tear strength. Also indentationmodulus linearly depends on the tear strength (Fig.5b).

As we mentioned in the experimental part it isphysically unreasonable to look for some relation be-tween microindentation parameters and parameterslike yield strength and tensile strength, because theybasically depend on macro defects. But when mea-sure the indentation characteristic there is one indirectindicator for existence of macro-defects- it is the re-sults scattering, i.e. the big error in measurements.Percentage errors vs. tensile strength are plotted in

(a) (b)

Fig. 5. All experimentally obtained microhardness characteristics (a) and indentation moduli (b) for the three investigatedsamples as a function of tear strength.

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Fig. 6. Relation between percentage error for all microin-dentation characteristics and tensile strength.

Fig. 7. Microindentation profiles.

Fig. 8. Creep curves.

Fig. 6. Sample 1 is the most inhomogeneous one andit has small tensile strength. Sample 2 is very ho-mogeneous and has the highest tensile strength. Thedependence has exponential decreasing character.

Fig. 7 shows microhardness profiles i.e. how thedynamic hardness changes in the depth of the speci-men. It should be emphasized that the hardness val-ues do not indicate the resultant hardness of the ma-terial exactly at this depth but incorporate the proper-ties of all surface layers up to that depth. Moreover atvery small depths (in this case less than about 4 µm)the hardness increases sharply, which by now has noprecise and unambiguous physical explanation. It’scalled “scale factor” or “size factor”. In this zonesome structural heterogeneity could be seen near tothe surface, but it is difficult to make some commentin this not well studied area.

The results for equilibrium swelling and effectivecrosslink density, which were determined by means ofDogadkin-Tarrasova’s method are shown in Table 2.It is observed almost linear dependence between ef-fective crosslink density and Shore hardness of inves-tigated samples from liquid silicone rubber.

Table 2. Values for equilibrium swelling and effectivecrosslink density, determined by Dogadkin-Tarrasova’smethod

Equilibrium Effective crosslinkswelling density

Sample Q n.10−19/cm3

1. SL7240 3.6 0.662. SL7250 3.1 0.913. SL7270 2.8 1.36

Fig. 8 shows the creep curves i.e. increasing of thedepth with time in constant load. The creep is morepronounced for sample 1 which has less crosslinkingand decreases with increasing of crosslinking. This islogical because denser lattice means bigger resistanceto time-dependent deformation. The creep curve forspecimen 1 is not smooth, confirming its low homo-geneity.

Comparison between experimental and numericalload-displacements curves is shown in Fig. 9 and inthe Table 3.

The distribution of equivalent Von Mises stress ininvestigated materials during the modelled process ofmicroindentation, as well as the distribution of theequivalent plastic strain after unloading are shown inFigs. 10-12. These figures show sink-in of material

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Table 3. Initial values for numerical simulations (taken from literature and manufacturer certificates) and material param-eter set, which gives the best fit with experimental results

Material Initial values Best fit

σp K Eit ν σp K Eit ν

(MPa) (-) (MPa) (-) (MPa) (-) (MPa) (-)

1 3.73 2.41 3.37 0.47 0.426 2.41 2.77 0.472 3.73 2.41 2.89 0.47 0.400 2.41 2.35 0.473 3.73 2.41 3.81 0.47 0.600 2.41 3.45 0.47

(a) (b) (c)

Fig. 9. Comparison between experimental and numerical load-displacement curves for material 1 (a), 2 (b) and 3 (c).

(a) (b)

Fig. 10. Distribution of equivalent Von Mises stress (a) and the equivalent plastic strain (b) in material 1.

(a) (b)


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(a) (b)


around indenter imprint, which could be taken intoaccount only by means of numerical simulations andcould be one of the reasons for explaining the differ-ence between experimental and numerical values ofelastic modulus.

CONCLUSIONS

In present work we have investigated the influenceof crosslinking on mechanical properties of three liq-uid silicon rubber samples. We established:• Increasing in the degree of crosslinking con-

firm that elasticity has no direct influence on themicrohardness characteristics because they arefully (Hit , HV*) or partially (DH, HMs) depen-dent from the plastic deformation (irreversibleslippage of the macromolecules);• Only creep curves are influenced by the

crosslinking. The creep is more pronounced forthe sample which has less crosslinking and de-creases with increasing of crosslinking;• There is a linear dependence between all micro-

hardness characteristics and indentation mod-ulus on one hand and tear strength and yieldstrength on the other hand. So this good cor-relation between these micro- and macro- char-acteristics indicates that both are affected in thesame way by the structural features of the rub-ber, but crosslinking is not a main decisive fac-tor;• The scattering of the results in microinden-

tation experiment, respectively magnitude ofthe experimental error can be regarded as anindirect indicator of the magnitude of tensilestrength.

Three series of numerical simulations were real-ized by means of the finite-element method and theresults were compared with experimental results frommicroindentation measurements. In this way, a trial-

error procedure was performed in order to obtain thematerial parameter set that gives the best fit to theexperimental data. The constitutive model used inthe finite-element model of microindentation processis based on the Von Mises yield criterion in combi-nation with a linear hardening law. Moreover fig-ures 10-12 show sink-in of material around indenterimprint, which could be taken into account only bymeans of numerical simulations. Numerical simula-tions by means of finite-element method, which werealized, gave us additional information about me-chanical properties of investigated materials, whichis impossible to obtain by means of microindentationexperiment only. Moreover this hybrid experimental-numerical approach save money and time for prepar-ing of test samples and realizing conventional exper-iments in order to obtain plastic behavior of investi-gated materials.

Acknowledgments. This paper was financiallysupported by Todor Kableshkov University of Trans-port (Project No 1620/22.04.2014).

REFERENCES

[1] B. A. Dogadkin, Z. N. Tarasova and A. S. Linkin,J. Vsesojusnovo Chimicheskova Obstestva ImeniMendeleeva (USSR) 13, 87 (1968).

[2] J. C. Hay, A. Bolshakov and G. M. Pharr, J. Mater.Res. 14, 2296–2305 (1999).

[3] G. M. Pharr and A. Bolshakov, J. Mater. Res. 17,2660–2671 (2002).

[4] A. Bolshakov and G.M. Pharr, “Inaccuracies in Sned-don’s solution for elastic indentation by a rigidcone and their implications for nanoindentation dataanalysis” in Proceedings of the Conference “SpringMeeting of the Materials Research Society” CONF-960401-16, Materials Research Society, San Fran-cisco, USA, May 1996.

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[5] MSC. Software Corporation, USA, MSC.MARCUser’s Guide, 2013.

[6] S. Cherneva and R. Iankov, “Investigation on theinfluence of the boundary conditions by simulationwith finite element method of microindentation pro-cess” in Proceedings of the Thirty Sixth Spring Con-ference of the Union of Bulgarian Mathematicians,

St. Konstantin & Elena Resort, Varna, Bulgaria,2007, pp. 197–202.

[7] Materials and Coatings for Medical Devices: Car-diovascular (book), ASM Materials for MedicalDevices Database Committee, ASM International,USA, November 2009, pp. 297-303.

ВЛИЯНИЕ НА СТЕПЕНТА НА ОМРЕЖВАНЕ ВЪРХУ МИКРОМЕХАНИЧНИТЕ ХАРАКТЕРИСТИКИ.СИМУЛАЦИЯ НА МИКРОИНДЕНТАЦИОННИЯ ПРОЦЕС

Г. Замфирова1, С. Чернева2, В. Гайдаров1, Т. Владкова3

1 ВТУ “Тодор Каблешков”, ул. “Гео Милев”№158, 1574 София, България2 Институт по механика, Българска академия на науките, ул. “Акад. Г. Бончев” блок 4, 1113 София, България

3 Химикотехнологичен и металургичен университет, бул. “Св. Кл. Охридски”№8, 1756 София, България

(Резюме)

Изследвани са образци от силиконов каучук с различна степен на омрежване чрез метода на контролирано проникванена индентора (Depth Sensing Indentation (DSI)). От експериментално получените индентационни криви са определени редицамикроиндентационни характеристики:

— микротвърдост по Викерс (HV*);— индентационна твърдост (Hit);— Мартенсова твърдост (HMs);— динамична твърдост (HMV);— модул на еластичност (Е);— отношение между еластичната и пластична компонента на деформацията;— криви на пълзене;— промяна на динамичната твърдост в дълбочина на образеца и др.Установено е, че степента на омрежване няма директно влияние върху микротвърдостните величини имодула, защото те са

свързани изцяло (Hit, HV*) или частично (HMV, HMs) с пластичните деформации. Но възвратимата деформация, която е сума отмигновенната Хуковска деформация и времезависимата вискоеластична деформация се влияят от гъстотата намрежата. Затовастепента на омрежване оказва почти линейно увеличение на работата, необходима за еластична деформация, и намаляване надълбочината на проникване при режим на пълзене.

Направена е симулация по метода на крайните елементи, чрез която са определени:— разпределение на еквивалентните напрежения на фон Мизес;— разпределение на пластичните деформации за сложното напрегнато състояние под индентора в натоварено състояние;— границата на пластично течение и нейната зависимост от степента на омрежване.Така този хибрид на експериментално-числен подход спестява голямо количество материал за изготвяне на опитни образ-

ци, средства и време за подготовката на пробите за тестване и реализиране на допълнителни експерименти.

Благодарности: Изследването е финансово подкрепено от ВТУ “Тодор Каблешков” (Проект№1509/25.05.2013).

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